The parameter space \(\mathcal {H}(c_{n,p})\)  contains a large collection of sparse covariance matrices and does not impose any constraint on the variances \(\sigma _{ii}\) , \(i=1, ..., p\) . The collection \(\mathcal {H}(c_{n,p})\)  contains some other commonly used classes of sparse covariance matrices in the literature, including an \(\ell _{q}\)  ball assumption \(\max _{i}\sum _{j=1}^{p}\left|\sigma _{ij}\right|^{q}\le s_{n,p}\) 
in Bickel and Levina [1]}, and a weak \(\ell _{q} \)  ball assumption \(\max _{1\le j\le p}\left\lbrace \left|\sigma _{j\left[ k\right] }\right|^{q}\right\rbrace  \le s_{n,p}/k \)  for each integer \(k\)  in Cai and Zhou [2]} where \(\left|\sigma _{j \left[ k\right] }\right|\)  is the \(k\) th largest entry in magnitude of the \(j\) th row \(\left( \sigma _{ij}\right) _{1\le i\le p}\) . See Cai et al. [3]} for more discussions.
